ABSTRACT In this paper, we study the k–Lucas numbers of arithmetic indexes of the form an+r , where n is a natural number and r is less than r. We prove a formula for the sum of these numbers and particularly the sums of the first k-Lucas numbers, and then for the even and the odd k-Lucas numbers. Later, we find the generating function of these numbers. Below we prove these same formulas for the alternated k-Lucas numbers. Then, we prove a relation between the k–Fibonacci numbers of indexes of the form 2rn and the k–Lucas numbers of indexes multiple of 4. Finally, we find a formula for the sum of the square of the k-Fibonacci even numbers by mean of the k–Lucas numbers.
Cite this paper
S. Falcon, "On the k–Lucas Numbers of Arithmetic Indexes," Applied Mathematics, Vol. 3 No. 10, 2012, pp. 1202-1206. doi: 10.4236/am.2012.310175.
 S. Falcon, “On the k-Lucas Numbers,” International Journal of Contemporary Mathematical Sciences, Vol. 6, No. 21, 2011, pp. 1039-1050
 S. Falcon and A. Plaza, “On the Fibonacci k-Numbers,” Chaos, Solitons & Fractals, Vol. 32, No. 5, 2007, pp. 1615-1624. doi:10.1016/j.chaos.2006.09.022
 S. Falcon and A. Plaza, “The k-Fibonacci Sequence and the Pascal 2-Triangle,” Chaos, Solitons & Fractals, Vol. 33, No. 1, 2007, pp. 38-49.
 S. Falcon and A. Plaza, “On k-Fibonacci Numbers of Arithmetic Indexes,” Applied Mathematics and Computation, Vol. 208, 2009, pp. 180-185